Analytical study of reaction diffusion Lengyel-Epstein system by generalized Riccati equation mapping method

In this study, the Lengyel-Epstein system is under investigation analytically. This is the reaction–diffusion system leading to the concentration of the inhibitor chlorite and the activator iodide, respectively. These concentrations of the inhibitor chlorite and the activator iodide are shown in the form of wave solutions. This is a reactionâ€ “diffusion model which considered for the first time analytically to explore the different abundant families of solitary wave structures. These exact solitary wave solutions are obtained by applying the generalized Riccati equation mapping method. The single and combined wave solutions are observed in shock, complex solitary-shock, shock singular, and periodic-singular forms. The rational solutions also emerged during the derivation. In the Lengyel-Epstein system, solitary waves can propagate at various rates. The harmony of the system’s diffusive and reactive effects frequently governs the speed of a single wave. Solitary waves can move at a variety of speeds depending on the factors and reaction kinetics. To show their physical behavior, the 3D and their corresponding contour plots are drawn for the different values of constants.


The generalized Riccati equation mapping method
The generalized Riccati equation mapping method is define in the following steps 40-43 .

Step I
Given nonlinear partial differential equation (NPDE) having independent variables x = (t, x, y, z) and depend- ent variable w where P is generally a polynomial function of its argument, and the subscripts of dependent variable denotes the partial derivatives.
Step II By using the wave transformation, Eq. (1) have the following ansatz (1) P(w, w t , w x , w y , w z , w xx , w zz , w xy , w tt • • • ) = 0, where η is a real function to be determined.Substituting Eqs. ( 2) into (1) then we get an ordinary differential equation (ODE) as

Step III
Suppose that the solution of Eq. ( 3) is in the polynomial form where a j are constants that are determine later and n is positive integer that is obtained by the help of balancing principle.The φ(η) represents the solution of the given generalized riccati equation.
where τ , ρ and χ are all real constants.Substituting the Eqs.( 4) with (5) into the regarding ODE and remove all the coefficients of φ will obtain a system of algebraic equations, from which we can get the parameters a j , = (j = 1, • • • , n) and η .Solving the algebraic equations, with the known solutions of Eq. ( 4), one can be easily obtain the non-travelling wave solutions to the NPDE Eq. (1).We can obtain the following twenty seven solutions to Eq. ( 3) such as Type 1: For ρ 2 − 4τ χ > 0 and ρτ = 0,(orρχ = 0) the solutions of Eq. ( 5) are, where A and B are two non zero real constants and satisfies B 2 − A 2 > 0.
(15) where c 1 is an arbitrary constant.

Application to Lengyel-Epstein reaction diffusion system
In this section, we investigate the analytical solutions of the Lengyel-Epstein system by using the generalized Riccati equation mapping method.Here we illustrate this couple system to achieve the analytical solution of the Lengyel-Epstein system 33,35,45,46   where u and v are the concentration of the inhibitor chlorite and the activator iodide, respectively.a , b and c are the constants.By the wave transformation Eq. ( 2) we convert the Eqs.( 33) and (34) onto the ODEs as follows Now, we suppose that the solution of Eqs.(35) and (36) as So, it is satisfies the auxiliary ODE as Now substituting the value of N by using the homogeneous balancing By finding the derivatives of the Eqs.(40) and (41) along with Eq. (39) and putting in the Eqs.(35) and (36), and get the system of equations.After solving the system of equation we get the solutions as follows; Case 1: Form the Eq. ( 35) we gain σ 0 = − . For the Eq. ( 36) Type 1: When ρ 2 − 4τ χ > 0 and ρχ = 0(orτ χ = 0) , we obtained hyperbolic solutions.Substituting the values of constants in Eqs.(40) and (41) and by the help of general solution that are mentioned in methodology we obtained the different form of solutions.
By putting constant values along with Eq. ( 18) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary solutions of Eqs. ( 33) and ( 34) such as By putting constant values along with Eq. ( 19) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as By putting constant values along with Eq. ( 20) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as Vol.:(0123456789) www.nature.com/scientificreports/By putting constant values along with Eq. ( 21) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as By putting constant values along with Eq. ( 22) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as By putting constant values along with Eq. ( 23) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as www.nature.com/scientificreports/By putting constant values along with Eq. ( 24) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as where G and H are two non-zero real constants and satisfies G 2 − H 2 > 0.
By putting constant values along with Eq. ( 25) and wave transformation Eq. ( 2) in the Eqs.(40) and (41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as where By putting constant values along with Eq. ( 26) and wave transformation Eq. ( 2) in the Eqs.( 40) and ( 41) we get the solitary wave solutions of Eqs. ( 33) and ( 34) such as

Graphical behaviors
In this section, we discuss the graphical behavior of the solutions that are successfully obtained by using the GREM method for the reaction-diffusion Lengyel-Epstein system.In summary, the GREM technique is a useful tool to obtain the exact solitary wave solutions and control theory, especially for linear time-invariant systems.Its applicability to a wide variety of control issues is constrained by the linearity assumption, complexity, and limits mentioned earlier

Conclusions
In this study, we find the analytical wave solutions for the Lengyel-Epstein reaction-diffusion system.The reaction-diffusion The Lengyel-Epstein model represents the concentration of the inhibitor chlorite and the activator iodide, respectively.These concentrations of the inhibitor chlorite and the activator iodide are shown in the form Reports | (2023) 13:20033 | https://doi.org/10.1038/s41598-023-47207-4
. When selecting whether to adopt GREM or look into other control approaches, engineers should take into account the unique characteristics of their systems and the issue at hand.Many physical significances are explained by sketching some three-dimensional diagrams and their corresponding contours for the acquired solutions.These figures give us a better understanding of the behavior of these solutions.The different solutions are plotted in 3D and their corresponding contour representations on the MATHEMATICA 11.1 for the different values of constants.These results are very helpful in the dynamic study of this chemical reaction model.The Figs. 1, 2, 3 and 4 show the kink type soliton behavior for the inhibitor chlorite using the range of space and temporal parameters [− 10,10] and [− 2,2] respectively.The Figs. 5, 6, 7, 8 and 9 show the solitary wave behaviors for the range of space and temporal parameters [− 1,1].The Figs. 10 and 11 are the lump